In our previous article, “On the Cesàro operator on the Hardy space in the upper half-plane” (arXiv: 2405. 19627), we proved that the Cesàro operator on the Hardy space H 2 (C +) H² (C_+) in the upper half-plane is the sum of the identity operator and a unitary operator. In this article we investigate properties of that unitary operator. We explicitly determine its resolution of identity and show that this unitary operator is unitarily equivalent to the multiplication by a rational unimodular on the real axis function in the space of Mellin transforms of functions from H 2 (C +) H² (C_+). In particular we deduce that the Cesàro operator on H 2 (C +) H² (C_+) is cyclic.
Andreev et al. (Wed,) studied this question.